On 2/19/2013 6:29 AM, Shmuel (Seymour J.) Metz wrote:> In <x_-dnZNsYePggrzMnZ2dnUVZ_rydnZ2d@giganews.com>, on 02/17/2013> at 12:20 PM, fom <fomJUNK@nyms.net> said:>>> This is not how I understand mathematics.>> There are two very different issues; structure and presentation. It is> cumbersome to always write proofs out in full, so working> mathematicians use a set of informal shorthand notations.>

Of course. And, I would not generally insist that any of it bealtered (except for one course in recursive function theorywhose professor worked without a book and filled the blackboardsfaster than anyone could transcribe the material...).

The problem arises elsewhere.

>> Almost every reputable mathematics department is>> giving courses in "mathematical logic," presumably>> based on this received paradigm.>> I believe that such courses fo into more than just the logical> machinery needed in Mathematics, and that they cover, e.g.,> independence, consistency, models.>

They do. But, the nature of identity is treated casually.It is that subject matter over which I stumbled. Thefirst hint I had that what I had been struggling with was, infact, legitimate came when I found this paper:

I did not discover that paper until 2003 or 2004. I had already formulated an alternative set of axioms for the treatment ofidentity in Zermelo-Fraenkel set theory. My first attempt atwriting a paper in 1994 had been dashed. I had written it in aformat called WordPerfect because I did not know about TeX.Then, I taught myself to administer a Linux platform (minimally)so that I could get a LaTeX installation. Of course, while doingthat, the MiTeX installation became available for Microsoftplatforms.

The second paper was submitted to "The Journal of Symbolic Logic".I do not believe it had been reviewed at all. The day aftersubmission, I received a confirmation email. It addressed me as"professor." I had been polite enough to inform them of theirerror. Within 30 minutes I received an angry rejection aboutwasting their time.

There had been an oversight in that paper (I hesitate to call itan error because it remains a different modal possibility.). Itis corrected now because of someone who had been kind enough torespond several years ago. It had not been a friendly response.However, it identified the error. The only supportive comment Iever received from anyone came from Wilfred Hodges who suggestedthat I should look at the work from the Polish school.

While I am not an expert on the pursuit of "quantum logic" inthe sense of what those lattice theorists who wrote the paperat the link above are pursuing, any computational model orsyntactic model with truth-functional capability will requiremore than what they have accomplished.

In the manipulation of syntax in those other posts I had linkedis a characterization of truth-functional behavior not based onspecific representation by truth tables. It is not based onBoolean algebra. In fact, if successful in its intent, itcharacterizes semantics relative to a lattice mapping from thefree DeMorgan lattice on one generator into the ortholatticeO_6 mentioned in the paper above.

>> But, in the "logical" sense,>> 1.000... = 0.999...>> is merely a stipulation of syntactic equality>> between distinct inscriptions that is prior>> to any mathematical discourse.>> There is no syntactic identity there. There is, instead, an identity> based on a specific[1] definition of the notation and a specific set> of axioms.>

Correct. My "stipulation of syntactic equality" is the same asyour "definition of notation".

The phrase "syntactic equality" is being taken directly from Carnap.

When I had been badly flamed quite some time ago, the flamer wouldkeep insisting that there was no problem with identity -- that, infact, any apparent problem is trivially addressed with term modelswhere the "objects" of the model are merely the equivalence classesof terms assigned equality by prior stipulation.

The difficulty here is that "mere" has nothing to do with it.

One has a set of symbols. One has a partition on that set ofsymbols. And, if one is claiming a homomorphism, then one islooking at the uniqueness of an induced quotient topology on theidentification space/decomposition space/quotient space.Individuals who are trained outside of mathematics do notrecognize the situation in this manner. In fact, when theyhear mathematicians use the word "identify" they often respondwith Russell's critical reference of Dedekind concerning"honest toil."

Along the same lines, a partition of the domain into equivalenceclasses has a lattice representation. Technically, such latticesare atomistic lattices having the atomic covering property:

if p and q are atoms and if b/\p=0 thenp<=b\/q implies q<=b\/p

They are more commonly referred to as matroid lattices and havea principal origin in the investigation of linearly dependentand linearly independent sets of vectors where the numerical coefficients have been ignored.

Needless to say, the axiom of choice and the generalizedcontinuum hypothesis come into play here.

Wittgenstein insisted on a form of identity wherein everydistinct symbol corresponded to a distinct object. We mightrefer to that as canonical naming. If, in the formation ofthe quotient space/quotient model above, one chooses insteadto use a map to a canonical name within each equivalence class,then separation properties of the discrete topology thatcharacterize the bottom of the partition lattice are retainedin moving up a chain through the lattice. This is a naturalway to retain the syntactic distinction of symbol shape.

Now within any equivalence class of symbols, distinguishingone at the expense of the others may be taken as fixing adirection on the edges of a complete graph such that thesymbol chosen as a canonical name is the only symbol forwhich every edge is outwardly directed.

Next, if one takes this as a prior constraint in relationto well-orderings, then the choice functions for the setare partitioned into those for which the chosen elementis first and those for which it is not. Among those forwhich it is first, one may now consider the labeling problemwhereby the ordinals correspond to the labels and thethe directed edges of the complete graph emanating fromthe canonical name are taken to be constraints.

In finite circumstances, path consistent labelings for acomplete graph reduce to path consistency along everytriple. Of course, to overlap triples sequentiallyis to overlap the pairs. This is, perhaps, one interpretationof Peano's axiom

(ae|N /\ be|N) -> (a=b <-> (a+1=b+1))

In addition, along the lines of the complete graph mentionedearlier, one must organize the triples so that they areoverlapping. Among all of my senseless syntax, you willfind the use of Steiner Quadruple Systems. Such blockdesigns are configurations wherein every triple is uniquelyrepresented in blocks of size 4. Because the 2-dimensionalflats of every n-dimensional affine space over the Galoisfield of order 2 form an SQS, an SQS can be formed forevery set of symbols with order 2^n, n>2.

For a long time, I did not know how to get the triplestopologically. Primarily, I had been focused on definitedescriptions and how Cantor's intersection theoremreflected Leibniz' actual statements better than Russell'slogicist representation. But, my attention turned toHausdorff spaces when I realized that unary negationfacilitated organizing the formulas of first order logicinto a minimal Hausdorff topology, provided one alsoaugmented the set with the Fregean notions of "the True"and "the False".

While looking at the connectedness properties of Peanospaces two days ago, I stumbled on Problem 28A in"General Topology" by Willard. The example discusseshow to form an indecomposable continua on 3 points.I am now looking at what I had been doing with theintersection theorem in terms of compact, connectedHausdorff spaces.

No. There is nothing "mere" about my understandingof these matters. And, you are correct. There areaxioms involved: